Table Of ContentRevisiting Ulysses Observations of Interstellar Helium
Brian E. Wood1, Hans-Reinhard Mu¨ller2, Manfred Witte3
5 ABSTRACT
1
0
2
We report the results of a comprehensive reanalysis of Ulysses observations of
n
a interstellar He atoms flowing through the solar system, the goal being to reassess
J
the interstellar He flow vector and to search for evidence of variability in this
2
1 vector. We find no evidence that the He beam seen by Ulysses changes at all
from 1994–2007. The direction of flow changes by no more than 0.3◦ and the
]
R speed by no more than 0.3 km s−1. Aglobal fit to all acceptable∼He beammaps
S ∼
. from 1994–2007 yields the following He flow parameters: VISM = 26.08 0.21
h ±
p km s−1, λ = 75.54 0.19◦, β = 5.44 0.24◦, and T = 7260 270 K; where λ
± − ± ±
-
o and β are the ecliptic longitude and latitude direction in J2000 coordinates. The
r flow vector is consistent with the original analysis of the Ulysses team, but our
t
s
a temperature is significantly higher. The higher temperature somewhat mitigates
[
a discrepancy that exists in the He flow parameters measured by Ulysses and
1
the Interstellar Boundary Explorer, but does not resolve it entirely. Using a
v
5 novel technique to infer photoionization loss rates directly from Ulysses data, we
2
7 estimate a density of nHe = 0.0196 0.0033 cm−3 in the interstellar medium.
±
2
0
. Subject headings: Sun: heliosphere — ISM: atoms
1
0
5
1
:
v 1. INTRODUCTION
i
X
r The global structure of the heliosphere is determined in large part by the local ISM
a
flow vector in the rest frame of the Sun. Older determinations of this vector, particularly
those based on measurements of interstellar He flowing through the inner heliosphere by
the GAS instrument on Ulysses, have recently been challenged by new measurements of the
He flow by the Interstellar Boundary Explorer (IBEX). The canonical Ulysses analysis of
1Naval Research Laboratory, Space Science Division, Washington, DC 20375, USA;
[email protected]
2Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755,USA
3Max-Planck-Institute for Solar System Research, Katlenburg-Lindau,Germany
– 2 –
Witte (2004) suggested the following He flow parameters: V = 26.3 0.4 km s−1, λ =
ISM
±
◦ ◦
75.4 0.5 , β = 5.2 0.2 , and T = 6300 340 K; where λ and β are the ecliptic longitude
± − ± ±
and latitude direction in J2000 coordinates. In contrast, IBEX data recently yielded the
following flow properties (Bzowski et al. 2012; M¨obius et al. 2012; McComas et al. 2012):
V = 23.2 0.3 km s−1, λ = 79.00 0.47◦, β = 4.98 0.21◦, and T = 6300 390 K.
ISM
± ± − ± ±
Significant inconsistencies exist between the Ulysses and IBEX measurements, particularly
for V and λ.
ISM
The lower V value of IBEX is of particular interest, as it may imply the nonexistence
ISM
of a bow shock around the heliosphere. The V = 23 27 km s−1 interstellar flow speed
ISM
−
happens to imply a fast magnetosonic Mach number of M 1, making the existence or
F
≈
nonexistence of a bow shock around the heliosphere very much an open question. Uncer-
tainties in the strength and orientation of the interstellar magnetic field, B , represent
ISM
one obstacle in determining whether a bow shock exists. The higher B is (and the more
ISM
perpendicular to the ISM flow), the lower M should be, and the less likely there is to be a
F
bow shock. But uncertainties in V are also important.
ISM
For many years the best assessments of ISM velocity suggested V 26 km s−1, not
ISM
≈
onlytheUlysses/GASmeasurements (Witte et al.1996; Witte2004) butmeasurements from
ISM absorption lines as well (V = 25.7 0.5 km s−1; Lallement & Bertin 1992). With
ISM
±
these relatively high values, heliospheric modelers favored M > 1, implying the existence of
F
abowshock. However, notonlyhasthenewIBEXmeasurement calledthisintoquestion, but
so has one reanalysis of ISM absorption line data (Redfield & Linsky 2008), which suggests
V = 23.84 0.90 km s−1. The lower V = 23 24 km s−1 values reported recently have
ISM ISM
± −
been enough for many to argue that M < 1 should be preferred (McComas et al. 2012;
F
Zieger et al. 2013). However, Scherer & Fichtner (2014) argue that including He+ density in
the calculation of Alfv´en speeds instead of just assuming a pure proton plasma would still
suggest M > 1 even if V 23 km s−1. On the other hand, Zank et al. (2013) note
F ISM
≈
that charge exchange processes may turn the bow shock into more of a “bow wave” even if
M > 1.
F
Inanycase, theexistence ornonexistence ofaheliospheric bowshock isoneissue driving
interest inwhether theHeflowvector ofUlyssesorthatofIBEXistobepreferred. Assuming
that the ISM cloud around the Sun is nonrigid, Gry & Jenkins (2014) have reanalyzed ISM
absorption line data and infer cloud kinematics more consistent with the Ulysses velocity
being valid close to the Sun. Ben-Jaffel et al. (2013), Lallement & Bertaux (2014), and
Vincent et al. (2014) have recently made additional arguments in favor of the older Ulysses
vector, implying thattheIBEXmeasurements must somehow beinerror. Frisch et al.(2013)
propose a very different solution. They propose that the local ISM flow actually varies with
– 3 –
time, and that these variations are responsible for the Ulysses/IBEX discrepancy. However,
observations of interstellar H from the Solar Wind Anisotropies instrument on the Solar and
Heliospheric Observatory and the Space Telescope Imaging Spectrograph instrument on the
Hubble Space Telescope do not provide support for a variable ISM flow vector (Vincent et al.
2014).
All this attention on the He flow vector provides motivation for taking another long look
at the old Ulysses/GAS data, in order to see if a new, independent analysis confirms the
Witte (2004) results, and to see if there is any evidence for He flow variability in the Ulysses
data that would support the Frisch et al. (2013) interpretation favoring a variable ISM flow
vector. Further justification is that while the Witte (2004) analysis considered data taken
from 1990–2002, later data acquired up until the end of the Ulysses mission in 2007 were not
considered.
Katushkina et al. (2014) have recently analyzed a couple of Ulysses He beam maps,
including one from 2007, demonstrating that the new IBEX He vector cannot reproduce
either map. They also note that the Witte (2004) vector might in principle be able to fit
the IBEX data if the temperature is allowed to be 9000 K, consistent with more extensive
∼
IBEX analyses (Bzowski et al. 2012; M¨obius et al. 2012; McComas et al. 2012). We here
present our own comprehensive reanalysis of the full Ulysses/GAS data set, providing a new
assessment of the He flow vector using analysis techniques developed independently from
both the original Ulysses/GAS team (Witte et al. 1996) and the IBEX team (Bzowski et al.
2012; M¨obius et al. 2012).
2. THE ULYSSES DATA
Launched in 1990 October, the primary mission of the Ulysses spacecraft was to study
various interplanetary constituents like the solar wind, magnetic fields, radio waves, dust,
solar andgalacticcosmic rays, etc. outside theecliptic planein anorbit passing over thesolar
poles (Wenzel et al. 1992; Marsden 2001). This orbit was accomplished with a gravitational
assist from Jupiter in 1992 February, leading to an orbit nearly perpendicular to the ecliptic
plane, with an aphelion near Jupiter’s distance of 5 AU, and a perihelion near 1 AU. Ulysses
continued to make observations from this orbit until the mission ended in late 2007. The
Ulysses/GAS instrument, described in detail by Witte et al. (1992), was designed primarily
to study interstellar neutral He atoms flowing through the inner heliosphere. Neutral helium
provides the best diagnostic of the undisturbed ISM flow due to the high abundance of
helium, and due to the fact that helium is not greatly affected by charge exchange during
its journey through the outer heliosphere, in contrast to hydrogen and oxygen, for example.
– 4 –
The first He observations were made by Ulysses/GAS during its cruise phase out to
Jupiter (1990–1992). These data were first studied by Witte et al. (1993). However, in this
paper we will focus on data taken during the operational phase of the Ulysses mission, with
Ulysses in its polar orbit after the Jupiter encounter. These He data are generally of higher
quality than the cruise phase data (Witte 2004), with better sampling of the He beam.
Additional advantages of the operations phase data will become very clear below.
Successful detection of interstellar He required that the inflow velocity be high enough
for the He atoms to exceed the particle detection threshold of the GAS instrument. With the
Ulysses orbital plane being basically perpendicular to the ISM flow vector, the detectability
of He depended mostly on the spacecraft velocity. Only when Ulysses was in the near-Sun
part of its orbit was the spacecraft traveling rapidly enough so that the incoming He had
sufficient energy to exceed the detection threshold. Thus, the Ulysses/GAS He observations
were confined to the three fast latitude scans made by Ulysses, in the periods 1994–1996,
2000–2002, and 2006–2007. These are the three epochs that we will focus on here. Witte
(2004) considered the first two epochs of observations, but not the last epoch.
The GAS device works essentially like a pin-hole camera with a single-element detector,
so observing the He beam requires making many different exposures in different directions to
fully map out the count rate distribution. The Ulysses spin axis, which was always pointed
towards Earth, defined how this was done. Ulysses had a rotation period of 12 s. During
this period, the GAS instrument would observe at a particular elevation angle, measured
relative to the spin axis, mapping out a ring on the celestial sphere. Detected counts were
◦
accumulatedin32bins, whichcoulduniformlycoverthefull360 rotation; ormorecommonly
◦ ◦ ◦
cover smaller arcs of 90 , 45 , or 22.5 . After a period of typically 68 min, the elevation angle
◦ ◦ ◦ ◦
scanned by GAS would be changed by a step of 1 , 2 , 4 , or 8 ; allowing another arc to be
observed. Over the course of 2 3 days, typically, a full scan across the He beam would be
−
completed, and a new map begun.
TheGASinstrumentactuallyutilizedtwodifferentchannelswithindependent detectors,
one with a wide field of view (WFOV) and one with a narrow field of view (NFOV), and
both channels were exposed simultaneously. Over the course of the three fast latitude scans
mentioned above, Ulysses obtained 400 WFOV maps and 400 NFOV maps. We will
∼ ∼
be working almost exclusively with the WFOV data here, which have higher signal-to-noise
(S/N) than the NFOV maps.
Figure 1 shows a typical WFOV map, from 2001 January 24. The GAS instrument
suffers from some sensitivity to UV emission even when in its particle detection mode
(Witte et al. 1996), so some hot stars are visible in the map, in addition to the broad He
beam. The stars can be inconvenient if blended with the He beam, but are actually useful
– 5 –
as calibrators of pointing accuracy otherwise. In this article we will in fact be using a re-
cent recalibration of the Ulysses/GAS data set making extensive use of the stars as pointing
calibrators, which is described in more detail by Bzowski et al. (2014). In an end-to-end
test, the calibration of the direction determination was adjusted so that the positions of
stars observed by the instrument coincided with their astronomical position. Based on many
more star observations from the whole mission a minor further adjustment of a few tenth of
a degree was required and subsequently all He data were reprocessed in 2013 with this new
calibration. This recalibration provides yet another motivation for reanalyzing these data,
though the change in the data is modest. Note that although the archival Ulysses data, and
its original analyses reported in the Witte et al. series of papers, are in B1950 coordinates,
the numbers and figures we provide in this article will be in the more modern J2000 system.
A final distinction to be made is that we will only be considering observations of the
so-called “direct beam” formed by He atoms that flow directly to the detector (with modest
gravitational deflection), as opposed to the “indirect beam” created by particles that go
around the Sun and are gravitationally redirected back towards the detector, arriving from
a very different direction than the direct beam atoms. The indirect beam is much fainter
than the direct beam due to massive photoionization and electron-impact ionization losses
that occur when these particles pass close to the Sun. Ulysses has nevertheless detected the
indirect beam in observations from 1995 (Witte 2004), but we will not be considering those
data here.
3. EMPIRICAL ANALYSIS
Rather than immediately try to extract a He flow vector from the data, which requires a
somewhat complex forwardmodeling analysis (see Sections 4 and 5), we find it worthwhile to
first make some purely empirical investigations of the beam. The primary motivation for this
is to look for any evidence that the beam is changing at all during the 1994-2007time period.
We quantify the properties of the beam using 2-D Gaussian fits to the beam, as mapped
into ecliptic coordinates (as in Figure 1). Note that there is no physical reason the beam
should be precisely Gaussian, but in practice the beams observed by Ulysses are reasonably
Gaussian in shape, so this is a practical way to quantify them. An example Gaussian fit is
shown in Figure 2. There are six free parameters of such a fit: the central longitude and
latitude, the latitudinal and longitudinal widths [specifically the full-width-at-half-maximum
(FWHM)], the count rate amplitude at beam center, and a background count rate; where
we assume a flat background underneath the He beam.
In the following discussion, we make use of an orbital phase for Ulysses, φ, based on the
– 6 –
angle defined by the positions of the spacecraft and the Sun, and the location of the ecliptic
plane crossing near aphelion. In this system, the ecliptic plane crossing near aphelion at
◦ ◦ ◦ ◦
ecliptic coordinates (λ,β)=(157.4 ,0 ) is at φ = 0.0, and that near perihelion at (337.4 ,0 )
is at φ = 0.5. The φ = 0.25 and φ = 0.75 orbital phases correspond to the near south pole
and near north pole locations (relative to the Sun), so Ulysses is below the ecliptic at φ < 0.5
and above it at φ > 0.5. (Ulysses travels from south to north during its fast latitude scans.)
The Gaussian fits are performed using a semi-automated procedure, but all the fits are
visually verified. Many beam maps end up being excluded for various reasons. At φ < 0.2
and φ > 0.9 the data are simply deemed too noisy. Observations with φ < 0.26 are excluded
due to contamination from a star (specifically the rightmost star seen in Figure 1). Some
maps are excluded for having a background that is not flat, possibly due to solar activity at
that time. The most common reason for exclusion is simply that some scans did not fully
cover the He beam. We only want to consider beam maps where the scan fully encompasses
the He beam. Ultimately, fits are accepted for 234 NFOV maps and 238 WFOV maps,
including the map in Figure 1. As mentioned above, we will be focusing on the higher S/N
WFOV data, but we always do verify that results are not different when considering the
NFOV data instead.
The Gaussian fit parameters of the WFOV maps are plotted in Figures 3 and 4. Fig-
ure3ashows thattheHebeamtracesoutahorseshoepatternintheskyduringafastlatitude
scan. Performing polynomial fits to longitude and latitude as a function of orbital phase al-
lows us to define the average beam location illustrated in Figure 3a. We will emphasize
below how the shape of this horseshoe is a powerful diagnostic for the He flow parameters.
Figure 3a demonstrates that basically the same horseshoe pattern is made during all three
fast latitude scans, showing little indication for variation in beam location. Figures 3b-d
provide a better indication for the deviations of beam position about the average. The scat-
ter of the data points is indicative of the measurement uncertainties, but some systematic
deviations are discernible despite the scatter. For example, the beam longitude appears to
be lower during epoch 3 by a few tenths of a degree. The beam latitude is also a few tenths
of a degree lower during epoch 3, but only for φ = 0.5 0.7. The latitudes observed during
−
epochs 1 and 2 are discrepant for φ = 0.75 0.85.
−
Of particular interest are longitude variations, as it is in longitude that the IBEX and
Ulysses flow vectors are in disagreement, and it is in longitude that Frisch et al. (2013)
claim evidence for a systematic increase with time. There is, however, no evidence for any
increase in longitude during the Ulysses era, with the epoch 3 data suggesting if anything
a slight decrease, as noted above. The average discrepancy in Figure 3d is 0.28◦. This can
be interpreted as an estimate of the pointing uncertainties inherent in the Ulysses data,
– 7 –
consistent with expectations based on the pointing calibration done using stars (Witte 2004;
Bzowski et al. 2014).
◦
In any case, the longitude variations in Figure 3b are much smaller than the 3.6 dis-
crepancy in longitude between the IBEX and Ulysses He flow vectors. We conclude that
there is no indication from the Ulysses data to support the notion that He flow variability is
responsible for the IBEX/Ulysses discrepancy. Furthermore, given that we now know that
the 2007 Ulysses data are not more consistent with the IBEX results than the older data,
there is only a 2 year time gap between the Ulysses and IBEX eras for the He flow to have
changed so significantly.
Figures 4c-d also demonstrate that there is no evidence for any significant change in
the He beam shape from 1994–2007. The latitudinal and longitudinal beam widths change
with orbital phase, but the same sort of variation is observed during all three fast latitude
scans. Figures 4a-b show that the three epochs do, however, exhibit noticeable differences
in background and beam amplitude. The Ulysses/GAS background is a complicated mix of
solar and cosmic sources, and we will not discuss it much here, other than to note that both
orbital phase and epochal variations are indicated by Figure 4b.
As for the beam count rate amplitude in Figure 4a, it naturally peaks near φ = 0.5
during each fast latitude scan, as that is when the spacecraft is moving fastest. The different
count rate levels observed during the three epochs are most naturally interpreted as being
indicative of variations in the solar EUV photoionization rate. It makes perfect sense for
epoch 2 (2000–2002) to have the lowest He fluxes, as this is a time near solar maximum,
with high photoionization rates and therefore substantial losses of neutral He due to pho-
toionization. In contrast, epochs 1 and 3 are both near solar minimum, thereby minimizing
photoionizationlosses. Wewill returntotherelatedissues ofphotoionizationandHedensity,
n , in Section 5.
He
4. MEASURING THE ULYSSES He FLOW VECTOR
The simple quantification of the He beam properties provided by the Gaussian fitting
in the previous section is not helpful for actually inferring a He flow vector from He beam
maps, which instead requires a forward modeling approach. In this procedure, a Maxwellian
He velocity distribution is assumed at infinity, characterized by five parameters (n , V ,
He ISM
λ, β, T), and then this Maxwellian is propagated to the position of the spacecraft. The
distribution will be primarily affected by the Sun’s gravitational influence. Loss processes
canalsobeaccountedforinthisforwardmodeling, themostimportantbeingphotoionization
– 8 –
by solar EUV photons. We will initially ignore photoionization in our forward modeling, but
we will consider it later to see if it changes our inferred He flow parameters.
The forward modeling method utilized in the original Ulysses/GAS team analyses is de-
scribed by Banaszkiewicz et al. (1996). Our forward modeling routine, which is described in
detail elsewhere (Mu¨ller & Cohen 2012; Mu¨ller 2012; Mu¨ller et al. 2013), involves a rather
different approach to the problem than the usual method involving numerical integration
from infinity. In our computations, quantities that are conserved along a trajectory, such
as total energy, angular momentum, eccentricity, and direction of perihelion, are used to
determine in one algebraic step the neutral He velocity at a desired point in space when
its velocity at infinity is given. Conversely, when both location and velocity of a He atom
are given, the conservation equations dictate a unique corresponding velocity at infinity.
The entire phase space distribution, F(v), can be calculated at a desired point in the inner
heliosphere by decomposing the problem into a family of trajectories with different veloci-
ties (different conserved quantities) that all converge at the chosen point. By assuming a
thermal Maxwellian as the original distribution at infinity, each of those trajectories can
then be assigned its corresponding phase space density, yielding F(v) at the point of interest
(Mu¨ller & Cohen 2012; Mu¨ller 2012; Mu¨ller et al. 2013).
Computing count rates in a given direction involves first transforming F(v) into the
spacecraft rest frame and converting into a coordinate system defined around the view direc-
tion of interest, resulting in F(V ,θ,ψ), where V is the speed relative to the spacecraft, θ is
s s
the poloidal angle with respect to the view direction, and ψ is the azimuthal angle. With a
volume element “d3v = V2sinθ dV dψ dθ”, the predicted count rate can then be expressed
s s
as
C = A V D (V ) G(θ) F(V ,θ,ψ) d3v, (1)
s e s s
Z
where A = 0.0908 cm2 is the effective area, D (V ) is the detection efficiency, and G(θ)
e s
is the detector geometry function. Figure 5 shows our digitized versions of G(θ) and D ,
e
approximated from Figures 1 and 3 of Banaszkiewicz et al. (1996), with the latter expressed
as a function of incoming particle energy, E = 1mV2, with m being the mass of the He
s 2 s
atom. For a given elevation angle, particles in the azimuthal bins described in Section 2 are
observed while the spacecraft is spinning, so there is “scan smoothing” along the azimuthal
direction. WeaccountforthisbyaveragingC valuescomputedatpixelcenter andat ∆α/2,
±
where ∆α is the azimuthal angular bin size.
By computing C for all n observed directions, we compute a synthetic He beam map,
C (for i=1,n), for comparison with the real one. This synthetic beam is multiplied by
i
a normalization factor, N, in order to best match the count rates of the observed beam.
′
This amounts to a revision of the assumed n value to n = N n . In making this
He He × He
– 9 –
normalizationwe areineffect replacing n with N asthe truedensity-related freeparameter
He
of our fit. We also have to account for the background count rate, B. As we did in the
Gaussian fitting, we simply assume a flat background. Thus, the new predicted count rates
are
′
C = N C +B. (2)
i × i
We perform a χ2 minimization (e.g., Bevington & Robinson 1992) to determine the N and
B values that lead to the best fit to the data. The χ2 value of this best fit represents our
measure of merit for this particular synthetic He beam map, which can be expressed as
n ′ 2
S C
χ2 = i − i , (3)
(cid:18) σ (cid:19)
Xi=1 i
′ ′
where C = C [N,B,C (V ,λ,β,T)] are the predicted count rates, S are the observed
i i i ISM i
count rates, σ their Poissonian uncertainties, and n the number of points considered in the
i
He beam map. We then have to determine what He flow parameters (V , λ, β, and T)
ISM
truly minimize χ2, representing the best fit to the data.
The standard approach for determining a set of best-fit parameters is to start with a
guess for the parameters, compute χ2, and then use a χ2 minimization routine to refine
the parameters to reduce χ2 until ultimately the true χ2 minimum is reached. If ν is the
number of degrees of freedom of the fit (the number of data points minus the number of free
parameters), then the reduced chi-squared is defined as χ2 = χ2/ν, which should be 1
ν ∼
for a good fit. We use the Marquardt method described by Press et al. (1989) as our χ2
minimization routine.
Figure 6a shows a WFOV map from 1995 August 20, and Figure 6b-c shows a fit to
this map, with the Witte (2004) vector used as our initial guess for the fit parameters. It
is a simple matter for the particle tracking code mentioned above (Mu¨ller & Cohen 2012;
Mu¨ller 2012) to compute the expected beam location based on the Witte (2004) vector, if
we assume T = 0 K (meaning all He atoms have the exact same velocity and follow the
exact same trajectory). We use this prediction to define the map region to be fitted. In
◦
particular, we only consider data points within 25 of this location (white dots in the figure),
which encompasses enough of the region beyond the He beam to estimate the background.
Figure 6b shows the synthetic beam map of the best fit, and Figure 6c explicitly plots
observed versus predicted count rates, with a data point plotted for each of the white dots
in Figure 6b. The He flow vector parameters listed in Figure 6c are gratifyingly close to the
Witte (2004) vector, and the χ2 = 1.06934 value indicates an excellent quality fit.
ν
Unfortunately, the best-fit parameters that we infer are actually highly dependent on
our initial guesses for the parameters. This is shown explicitly in a second fit to the 1995 Au-
– 10 –
gust 20 data in Figure 6d-e. With an initial guess very different from the Witte (2004) vector
used in Figure 6b-c, we end up with a fit to the data with very different parameters. The
parameters listed in Figure 6e are very far from all modern estimates of the local ISM flow
vector, so we know they are wrong. But these parameters lead to a fit that is just as good as
the more plausible fit in Figure 6b, with a χ2 = 1.05731 value that is actually slightly lower
ν
even.
This exercise emphasizes just how difficult it is to find a unique He flow vector from
a single He beam map. A single map does not sufficiently constrain the problem. There
are significant degeneracies among the fit parameters, leading to long diagonal troughs in
χ2 space where it is very hard to find the true χ2 minimum along the trough. This is true
for the analysis of IBEX data as well (see Figure 22 in Bzowski et al. 2012). The most
basic degeneracy is between flow direction and flow speed, which can be explained by the
following question: If a beam is seen from a given direction, is that due to a fast beam
coming from roughly that direction, or due to a slower beam that has been gravitationally
deflected into that direction? Since the He flow vector happens to lie near the ecliptic plane,
this degeneracy basically ends up mostly between V and λ (rather than β).
ISM
We argue here that these degeneracies can be effectively broken if Ulysses data from
different parts of the Ulysses orbit are fitted together, as opposed to fitting one beam map
at a time, as in Figure 6. The utility of this approach is illustrated by Figure 7. We had
previously emphasized how the He beam traces out a horseshoe pattern in the sky during a
fast latitude scan (see Figure 3a). Figure 7 shows the horseshoe pattern expected based on
the Witte (2004) vector (assuming T = 0 K), and also shows what happens to the horseshoe
when β, λ, and V are changed. When β and λ are changed the horseshoe shifts vertically
ISM
and horizontally, respectively. In contrast, V affects the size of the horseshoe, with higher
ISM
V yielding a smaller horseshoe. The crucial point is that V affects the horseshoe shape
ISM ISM
very differently from β and λ, thereby breaking the direction/velocity degeneracy noted
above. But this diagnostic power can only be properly considered if He beam maps from all
parts of the horseshoe are considered simultaneously in fitting the data; hence the global fit
approach.
The specific data used in this global fit approach are the 238 WFOV maps used in
the Gaussian fitting (see Section 3). The global fit is similar to the individual map fits in
Figure 6, but we are now fitting many more data points, naturally; 81,405 count rates within
the 238 maps, to be precise. In addition to the four He flow parameters of interest (V ,
ISM
λ, β, and T), there are also N and B parameters for each individual map, for a total of 480
